Optimum Receiver in AWGN (next)

Let us show now that r’(t) ↔ r is asufficient statistic, namely n’(t) does not add any further information to take the decision over the transmitted signal (symbol).

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN implemented with matched filters

Properties of the matched filter

Given a signal s(t) of duration T, the matched filter to s(t) is a filter with impulse response h(t) = s(T-t) of duration T and ≠ 0 in the interval (0,T)

Ex: Show that the output of the matched filter to s(t), when the input is s(t), is Rs(T-t) and, therefore, accounting from the properties of the autocorelation function is maximum in t =T and equal to Εs

Optimum decision : MAP criterion (next)

Optimum decision : MAP criterion (next)

Optimum decision : MAP criterion

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Optimum Receiver in AWGN (next)

Digital transmission over AWGN channel (next)

Example:QAM signalling (quadrature amplitude modulation)

um(t)=AmcgT(t)cos2πfct+AmsgT(t)sin2πfct

m=1,2,…,M

{Amc} and {Ams} are the sets of the amplitudes of the signals such that any binary k-sequence maps to a pair of different amplitudes